Ebrahim-Farrington Goodness of Fit test

Overview

The ebrahim.gof package implements the Ebrahim-Farrington goodness-of-fit test for logistic regression models. This test is particularly effective for binary data and sparse datasets, providing an improved alternative to the traditional Hosmer-Lemeshow test.

Key Features

• Ebrahim-Farrington Test: Simplified implementation for binary data with automatic grouping
• Original Farrington Test: Full implementation for grouped data
• Robust Performance: Particularly effective with sparse data and binary outcomes
• Easy to Use: Simple function interface similar to other goodness-of-fit tests
• Well Documented: Comprehensive documentation with examples

Installation

From GitHub (Development Version)

Copy and paste this in R or R-studio. ```r

Install devtools if you haven't already

if (!requireNamespace("devtools", quietly = TRUE)) { install.packages("devtools") }

Install ebrahim.gof from GitHub

devtools::install_github("ebrahimkhaled/ebrahim.gof") ```

From CRAN (Stable Version) (NOT AVAILABLE YET)

Another way to install the R-Libarary, but its not avaialbe yet. ```r

Will be available after CRAN submission

install.packages("ebrahim.gof") ```

Quick Start

```r library(ebrahim.gof)

Example with binary data

set.seed(123) n <- 500 x <- rnorm(n) linpred <- 0.5 + 1.2 * x prob <- 1 / (1 + exp(-linpred)) y <- rbinom(n, 1, prob)

Fit logistic regression

model <- glm(y ~ x, family = binomial()) predicted_probs <- fitted(model)

Perform Ebrahim-Farrington test

result <- ef.gof(y, predicted_probs, G = 10) print(result) ```

Main Functions

ef.gof()

The main function that performs the goodness-of-fit test:

r ef.gof(y, predicted_probs , G = 10, model = NULL, m = NULL)

Parameters: - y: Binary response vector (0/1) or success counts for grouped data - predicted_probs: Vector of predicted probabilities from logistic model - G: Number of groups for binary data (default: 10) - model: Optional glm object (required for original Farrington only, not for Ebrahim-Farrington test) - m: Optional vector of trial counts (for grouped data) (required for original Farrington only, not for Ebrahim-Farrington test)

Returns: A data frame with test name, test statistic, and p-value.

Examples

Example 1: Basic Usage with Binary Data

```r library(ebrahim.gof)

Simulate binary data

set.seed(42) n <- 1000 x1 <- rnorm(n) x2 <- rnorm(n) linpred <- -0.5 + 0.8 * x1 + 0.6 * x2 prob <- plogis(linpred) y <- rbinom(n, 1, prob)

Fit logistic regression

model <- glm(y ~ x1 + x2, family = binomial()) predicted_probs <- fitted(model)

Test goodness of fit

result <- ef.gof(y, predicted_probs, G = 10) print(result)

> Test TestStatistic pvalue

> 1 Ebrahim-Farrington -0.8944 0.8143

```

Example 2: Compare Different Group Numbers

```r

Test with different numbers of groups

results <- data.frame( Groups = c(4, 10, 20), Pvalue = c( ef.gof(y, predictedprobs, G = 4)$pvalue, ef.gof(y, predictedprobs, G = 10)$pvalue, ef.gof(y, predictedprobs, G = 20)$p_value ) ) print(results) ```

Example 3: Comparison with Hosmer-Lemeshow Test

```r library(ResourceSelection)

Ebrahim-Farrington test

efresult <- ef.gof(y, predictedprobs, G = 10)

Hosmer-Lemeshow test

hlresult <- hoslem.test(y, predictedprobs, g = 10)

Compare results

comparison <- data.frame( Test = c("Ebrahim-Farrington", "Hosmer-Lemeshow"), Pvalue = c(efresult$pvalue, hlresult$p.value) ) print(comparison) ```

Example 4: Power Analysis

```r

Function to simulate misspecified model

simulatepower <- function(n, betaquad = 0.1, n_sims = 100) { rejections <- 0

for (i in 1:nsims) { x <- runif(n, -2, 2) # True model has quadratic term linpredtrue <- 0 + x + betaquad * x^2 probtrue <- plogis(linpredtrue) y <- rbinom(n, 1, probtrue)

# Fit misspecified linear model
model_mis <- glm(y ~ x, family = binomial())
pred_probs <- fitted(model_mis)

# Test goodness of fit
test_result <- ef.gof(y, pred_probs, G = 10)

if (test_result$p_value < 0.05) {
  rejections <- rejections + 1
}

}

return(rejections / n_sims) }

Calculate power for different sample sizes

powerresults <- data.frame( n = c(100, 200, 500, 1000), power = sapply(c(100, 200, 500, 1000), simulatepower) ) print(power_results) ```

Methodology

The Ebrahim-Farrington test is based on Farrington's (1996) theoretical framework but simplified for practical implementation with binary data. The test uses a modified Pearson chi-square statistic:

For binary data with automatic grouping, the test statistic is:

Z_EF = (T_EF - (G - 2)) / sqrt(2(G - 2))

Where: - T_EF is the modified Pearson chi-square statistic - G is the number of groups - The test statistic follows a standard normal distribution under H₀

Advantages over Hosmer-Lemeshow Test

1. Better Power: More sensitive to model misspecification
2. Sparse Data Handling: Specifically designed for sparse data situations
3. Computational Efficiency: Simplified calculations for binary data
4. Theoretical Foundation: Based on rigorous asymptotic theory ## Superior Performance at G=10 Simulation results consistently demonstrate that the Ebrahim-Farrington test outperforms the Hosmer-Lemeshow test, even when the model misspecification is minimal—such as with a missing interaction or omitted quadratic term—when using G = 10 groups (Ebrahim, 2025).

Assympotitically Following the Standard Normal Distn

The following two figures illustrate that, under the null hypothesis, the Ebrahim-Farrington test statistic is asymptotically standard normal for both single-predictor and multiple-predictor logistic regression models. This property holds even in sparse data settings, confirming the theoretical foundation of the test and supporting its use for model assessment. (see (Ebrahim,2025))

• Figure 1: Empirical cumulative distribution function (CDF) of the Ebrahim-Farrington test statistic under the null for a single predictor, compared to the standard normal CDF.
• Figure 2: Empirical CDF for the test statistic under the null for a multiple independent predictors scenario, again compared to the standard normal.

These results demonstrate that the Ebrahim-Farrington test maintains the correct type I error rate and its statistic converges to the standard normal distribution as sample size increases, validating its asymptotic properties.

References

1. Farrington, C. P. (1996). On Assessing Goodness of Fit of Generalized Linear Models to Sparse Data. Journal of the Royal Statistical Society. Series B (Methodological), 58(2), 349-360.

2. Ebrahim, Khaled Ebrahim (2025). Goodness-of-Fits Tests and Calibration Machine Learning Algorithms for Logistic Regression Model with Sparse Data. Master's Thesis, Alexandria University.

3. Hosmer, D. W., & Lemeshow, S. (2000). Applied Logistic Regression, Second Edition. New York: Wiley.

Citation

If you use this package in your research, please cite:

Ebrahim, K. E. (2025). ebrahim.gof: Ebrahim-Farrington Goodness-of-Fit Test for Logistic Regression. R package version 1.0.0. https://github.com/ebrahimkhaled/ebrahim.gof

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

License

This project is licensed under the GPL-3 License - see the LICENSE file for details.

Author

Ebrahim Khaled Ebrahim
Alexandria University
Email: ebrahim.khaled@alexu.edu.eg

Acknowledgments

• Prof. Osama Abd ElAziz Hussien (Alexandria University) for supervision
• Dr. Ahmed El-Kotory (Alexandria University) for guidance and supervision
• The R community for continuous support and feedback